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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

All types implies torsion


Author: Walter Parry
Journal: Proc. Amer. Math. Soc. 110 (1990), 871-875
MSC: Primary 20K15
MathSciNet review: 1039537
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Abstract: We prove the following theorem. Given a positive integer $ n$ and a subset $ A$ of $ {{\mathbf{Z}}^n}$ with the following properties: (1) for every $ \left( {{a_1}, \ldots ,{a_n}} \right)$ in $ A$ the inequality $ \Sigma_{i = 1}^n {\left\vert {{a_i}} \right\vert} \geq 2$ holds, and (2) for every $ \left( {{x_1}, \ldots ,{x_n}} \right)$ in $ {{\mathbf{R}}^n}$ there exists an $ \left( {{a_1}, \ldots ,{a_n}} \right)$ in $ A$ with $ {a_i}{x_i} \geq 0$ for $ i = 1, \ldots ,n$, there exists a subset $ {A_0}$ of $ A$ such that $ {{\mathbf{Z}}^n}$ modulo the subgroup generated by $ {A_0}$ contains a nontrivial torsion element.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1039537-7
PII: S 0002-9939(1990)1039537-7
Article copyright: © Copyright 1990 American Mathematical Society